Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 11, pp. 1–20.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
QUASI-PERIODIC SOLUTIONS OF NONLINEAR BEAM
EQUATIONS WITH QUINTIC QUASI-PERIODIC
NONLINEARITIES
QIUJU TUO, JIANGUO SI
Abstract. In this article, we consider the one-dimensional nonlinear beam
equations with quasi-periodic quintic nonlinearities
utt + uxxxx + (B + εφ(t))u5 = 0
under periodic boundary conditions, where B is a positive constant, ε is a
small positive parameter, φ(t) is a real analytic quasi-periodic function in
t with frequency vector ω = (ω1 , ω2 , . . . , ωm ). It is proved that the above
equation admits many quasi-periodic solutions by KAM theory and partial
Birkhoff normal form.
1. Introduction and statement of main results
Recently, there has been a lot of publications concerning the dynamic behavior
for nonlinear beam equations by using different methods, see for instance, [1, 3, 8,
9, 11, 20, 27]. In these works, little is considered about quasi-periodic solutions of
this kind of equations. Significant results have been obtained with respect to quasiperiodic solutions of autonomous beam equations by KAM theory, see [13, 14, 15].
In particular, Liang and Geng [21] considered the existence of the quasi-periodic
solutions of completely resonant beam equations
utt + uxxxx + Bu3 = 0
(1.1)
with hinged boundary conditions with B = 1.
The works mentioned above do not non-autonomous include the case. More
recently, Wang [26] obtained the existence of quasi-periodic solutions for the nonautonomous beam equation
utt + uxxxx + µu + εg(ωt, x)u3 = 0
under hinged boundary conditions
u(t, 0) = uxx (t, 0) = u(t, π) = uxx (t, π) = 0.
2000 Mathematics Subject Classification. 35L05, 37K50, 58E05.
Key words and phrases. Infinite dimensional Hamiltonian systems; KAM theory;
beam equations; quasi-periodic solutions; invariant torus.
c
2015
Texas State University - San Marcos.
Submitted October 24, 2014. Published January 7, 2015.
1
(1.2)
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Q. TUO, J. SI
EJDE-2015/11
In this paper, we consider the existence of quasi-periodic solutions for nonlinear
beam equations with quintic quasi-periodic nonlinearities
utt + uxxxx + (B + εφ(t))u5 = 0,
(1.3)
subject to periodic boundary conditions
u(t, x) = u(t, x + 2π),
(1.4)
where B is a positive constant, ε is a small positive parameter, φ(t) is a real analytic
quasi-periodic function in t with frequency vector ω = (ω1 , ω2 . . . , ωm ). Equation
(1.3) can be regarded as a quasi-periodic perturbation (with perturbation term
εφ(t)u5 ) of the completely resonant nonlinear beam equation
utt + uxxxx + Bu5 = 0.
Firstly, we consider the existence quasi-periodic solutions of an ordinary differential equation with respect to the unknown function x(t),
ẍ + (B + εφ(t))x5 = 0.
(1.5)
Secondly, we obtain the nonlinear beam equation
¯ ξ,
¯ ε)v +
vtt + vxxxx + V1 (ω̃(ξ)t,
4
X
¯ ξ,
¯ ε)v k+1 = 0
εk Vk+1 (ω̃(ξ)t,
(1.6)
k=1
by letting u = u0 (t)+εv(x, t) in (1.3), here u0 (t) is a nonzero quasi-periodic solution
of (1.5) and Vk (k = 1, 2, . . . , 4) defined as in Section 3.
Thirdly, we construct the invariant tori or quasi-periodic solutions of (1.6) with
(1.4) by means of KAM theory. Finally, we prove that (1.3) with (1.4) have many
quasi-periodic solutions in the neighborhood of the quasi-periodic solutions to (1.5).
As in our previous works [25, 28, 30], the method used in this paper is based
on the infinite-dimensional KAM theory as developed by Kuksin [18] and Pöchel
[23]. Thus the main step is to reduce the equation to a setting where KAM theory
for PDE can be applied. We note that (1.6) is a nonlinear beam equation with
quasi-periodic potential and quasi-periodic nonlinearities, which needs to reduce
the linearized system of (1.6) to constant coefficients by a linear quasi-periodic
change of variables with the same basic frequencies as the initial system. However,
we cannot guarantee in general such reducibility. The strategy of the proof in this
paper is similar to the one in [25]. However, the details are quite different.
Now the reducibility problems of infinite-dimensional linear quasi-periodic systems, by KAM techniques, has become an active field of research. The first result
was obtained by Bambusi and Graffi [2], after that, Eliasson and Kuksin [10], Yuan
[29], Liu and Yuan [22], and Grébert and Thomann [16]. However, in general, the
reducibility of infinite-dimensional linear quasi-periodic systems remains open and
is very attracting.
For our purpose, we first introduce a hypothesis.
(H1) B > 0, φ(t) is a real analytic quasi-periodic function in t with frequency
vector ω = (ω1 , ω2 , . . . , ωm ), where ω ∈ DΛ ,
DΛ := {ω ∈ Rm : |hk, ωi| ≥ Λ|k|−(m+1) , 0 6= k ∈ Zm }
with Λ > 0.
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QUASI-PERIODIC SOLUTIONS
3
For γ > 0 we define the set
Aγ = {α ∈ R : |hk, ωi + lα| > γ(|k| + |l|)−(m+1) , for all 0 6= (k, l) ∈ Zm × Z}
with its complex neighborhood Aγ + h of radius h. The following theorem is the
main result of this article.
Theorem 1.1. Assume that (H1) is satisfied. For an arbitrary index set Nd =
{n1 , n2 , . . . , nd } ⊂ N, there is a small enough positive ε∗∗ such that for any 0 < ε <
ε∗∗ , there are the sets J ⊂ Jˆ ⊂ [π/T, 3π/T ] and Σε ⊂ Σ := DΛ × Aγ × [0, 1]d+1
with meas Jˆ > 0, meas J > 0 and meas (Σ \ Σε ) ≤ ε, such that for any ξ¯ ∈ J
¯ ξ˜0 L(ξ˜j )j∈N ) ∈ Σε , the nonlinear beam equation (1.3)-(1.4) possess
and (ω, α(ξ),
d
a solution of the form
u(t, x) = u0 (t) + εu1 (t, x) + o(ε),
q
q
X
2ξ˜j /π
q
cos j 4 + ε[V̂ ]t cos(jx),
u1 (t, x) =
4
j 4 + ε[V̂ ]
j∈{0}∪Nd
with frequency vector
b
¯ (ω̂j ){0}∪N ) ∈ Rm+d+2 ,
ω
b = (ω, α(ξ),
d
and V̂ as defined in Section 3.
(i) ω is the frequency vector of φ , while α, ω̂j are constructed in the proof, and
¯ ξ˜ = (ξ˜0 , (ξ˜j )j∈N ) ∈ Rd+1 . In particular,
are functions of ε and of parameters ξ,
d
q
¯ ε), µj (ε) = j 4 + ε[V̂ ] + O(ε5.5 ), j ∈ {0} ∪ Nd ,
ω̂j = µj (ε) + ε2 a(ξ,
˜ ε)| ≤ C;
where |aj (ξ,
¯ ε),
(ii) u0 (t) is a non-trivial solution of (1.5) depending on the parameters (ξ,
it is of size O(ε1/4 ) and of quasi-periodic with frequency (ω, α), and u1 (t, x) is a
solution of the linear equation
∂tt u1 + ∂xxxx u1 + ε[V̂ ]u1 = 0,
q
and is quasi-periodic with frequencies j 4 + ε[V̂ ], j ∈ {0} ∪ Nd .
(1.7)
Remark 1.2. Using the method of this paper we cannot expect an existence result
for quasi-periodic solutions u(t, x) of (1.3) with the same frequency ω as the data
of the problem. Actually, the quasi-periodic solutions obtained in Theorem 1.1
bifurcate from quasi-periodic solutions of the nonlinear ODE (1.5), and the solutions
have more frequencies than the data of the problem. That is because we need to
add extra parameters while one considers the existence of quasi-periodic solutions
for ODE (1.5) and PDE (1.6) by means of KAM theory respectively. The addition
of the extra parameters will cause solutions with additional frequencies. Thus,
the main aim of the work is the construction of solutions with also additional
frequencies, and the solutions could be viewed as the interaction between u0 (t),
which is the (ω, α)-quasi-periodic solutions of (1.5), and u1 (t, x), which is a solution
1/2
of the linear equation (1.7), and is quasi-periodic with frequencies j 4 + ε[V̂ ]
.
The result is still new even if the equation seems to be quite studied and wellunderstood.
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Q. TUO, J. SI
EJDE-2015/11
Remark 1.3. To avoid the double eigenvalues we restrict ourselves to choose an
even complete orthogonal basis φj (x) = √1π cos(jx), j ≥ 0. The solutions u(t, x)
constructed in Theorem 1.1 are even in space variable x.
2. Quasi-periodic solutions of a nonlinear ODE
In 2000, Bibikov [5] developed a KAM theorem for nearly integrable Hamiltonian
systems with one degree of freedom under the quasi-periodic perturbation:
∂H(r, ϕ, ωt, a)
,
∂ϕ
(2.1)
∂H(r, ϕ, ωt, a)
ϕ̇ = a +
∂r
with a one-dimensional parameter a. In this section, we apply the results in [5]
(See also [17]) to show the existence of the quasi-periodic solutions for the following
nonlinear ordinary differential equation with quasi-periodic coefficient,
ṙ = −
ẍ + (B + εφ(t))x5 = 0.
(2.2)
First we introduce the Bibikov’s lemma.
Lemma 2.1 ([5]). Suppose that H(r, ϕ, θ, a) is real analytic on
1
D = {(r, ϕ, ωt, a) : |r| < δ0 , | Im ϕ| < p0 , | Im θ| < p0 , a ∈ Aγ + γδ0 }
2
and satisfies
|H| < γpm+3
δ02 .
(2.3)
0
Then
(i) There exists a δ0∗ such that if δ0 < δ0∗ , then there exists a function a0 :, Aγ → R
and a change of variables
r = ρ + v(ρ, ψ, θ, α),
ϕ = ψ + u(ψ, θ, α),
α ∈ Aγ ,
that transforms system (2.1) with a = a0 (α) into a system
ρ̇ = B(ρ, ψ, θ, α),
ψ̇ = Ψ(ρ, ψ, θ, α)
that satisfies the condition
B(0, ψ, θ, α) =
∂B(0, ψ, θ, α)
= Ψ(0, ψ, θ, α) = 0.
∂ρ
(ii) meas(AKµ ) → µ, as K → 0+ , where I ⊂ R is a unit interval, µ > 0, and
AKµ = {α ∈ µI : |hk, ωi + lα| ≥ Kµ(|k| + |l|)−(m+1) , for all 0 6= (k, l) ∈ Zm × Z}.
Equation (2.2) is equivalent to the system
ẋ = −y,
ẏ = Bx5 + εφ(t)x5 .
(2.4)
Using Bibikov’s theorem, we have the following lemma.
Lemma 2.2. For any ω ∈ DΛ , there exists an ε∗ such that for any positive 0 < ε <
ε∗ and sufficiently small γ > 0 there exist a real analytic function a0 (α) : Aγ → R
and a set Jˆ ⊂ [π/T, 3π/T ] with meas Jˆ > 0, such that for α ∈ Aγ , ξ¯ ∈ Jˆ and
¯ ε) ∈ Qσ (ω̃) with
some σ > 0, Equation (2.2) has a quasi-periodic solution x(t, ξ,
1
¯ = (ω1 , ω2 , . . . , ωm , α(ξ))
¯ satisfying x(t, ξ,
¯ ε) = O(ε 4 ).
ω̃(ξ)
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QUASI-PERIODIC SOLUTIONS
5
¯ is obtained
Remark 2.3. ω = (ω1 , ω2 . . . , ωm ) is the frequency of φ(t) and α(ξ)
from Lemma 2.2, then the frequency of solution for (2.2) is dimension m + 1.
3. Hamiltonian formalism
From Lemma 2.2 we know that for every ε ∈ (0, ε∗ ) equation (2.2) has a non¯ ε) with frequency vector ω̃(ξ).
¯ Taking u =
trivial quasi-periodic solution u0 (t, ξ,
¯
u0 (t, ξ, ε) + εv(t, x) in (1.3), we get the equation
¯ ξ,
¯ ε)v +
vtt + vxxxx + V1 (ω̃(ξ)t,
4
X
¯ ξ,
¯ ε)v k+1 = 0,
εk Vk+1 (ω̃(ξ)t,
(3.1)
k=1
where
4
¯ ξ,
¯ ε) := 5 B + εφ(ωt)
b
¯ ξ,
¯ ε),
V1 (ω̃(ξ)t,
ū0 (ω̃(ξ)t,
b
¯ ξ,
¯ ε), k = 1, 2, 3, 4
¯ ξ,
¯ ε) := C k+1 B + εφ(ωt)
ū4−k
(ω̃(ξ)t,
Vk+1 (ω̃(ξ)t,
0
5
¯ and φb and ū0 are the shell
are quasi-periodic in time t with frequency vector ω̃(ξ),
functions of φ and u0 respectively. Let us write
¯ ε)
V̂ (θ, ξ,
(3.2)
4
√
b
= 5c B + εφ(ωt)
(ξ¯ + εI)1/4 C(ϕT )
¯ ∈ Tm+1 , and
with θ = ω̃(ξ)t
√ dI
d
¯ ε) = 5c B + εφ(ωt)
b
V̂ (θ, ξ,
(1 + ε ¯)C 4 (ϕT )
¯
dξ
dξ
√
dϕ + (ξ + εI)4C 3 (ϕT )C 0 (ϕT ) ¯ .
dξ
Therefore,
Z
1
¯ ε)dθ = 5cB ξC
¯ 4 (ϕ0 T ),
lim V̂ (θ, ξ,
ε→0
(2π)m+1 Tm+1 ε→0
Z
d
1
d
¯ ε)dθ = 5cBC 4 (ϕ0 T ).
¯ ε)] =
lim ¯[V̂ (θ, ξ,
lim ¯V̂ (θ, ξ,
m+1
ε→0 dξ
(2π)
Tm+1 ε→0 dξ
¯ ε)] =
lim [V̂ (θ, ξ,
Thus, there exists 0 < ε1 < ε∗ such that for any ε ∈ (0, ε1 ),
¯ ε)] > 5 cB ξC
¯ 4 (ϕ0 T ) := I1 > 0,
[V̂ (θ, ξ,
2
∂
¯ ε)] > 5 cBC 4 (ϕ0 T ) := I2 > 0.
[V̂ (θ, ξ,
¯
2
∂ξ
Let us write
¯ ε) := [V̂ (θ, ξ,
¯ ε)] + Ve (θ, ξ,
¯ ε)
V̂ (θ, ξ,
¯ ε)] = 0, and mε := ε[V̂ (θ, ξ,
¯ ε)] > 0. Then
with [Ve (θ, ξ,
(3.3)
(3.4)
¯ ε) = εV̂ (θ, ξ,
¯ ε) = mε + εVe (θ, ξ,
¯ ε).
V1 (θ, ξ,
Furthermore, we can show that
¯ ε) = O(ε)
Ve (θ, ξ,
as ε 1. In fact, from (3.2), we have
√
¯ ε) = 5c B + εφ(ωt)
b
V̂ (θ, ξ,
(ξ + εI)C 4 (ϕT ),
(3.5)
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Q. TUO, J. SI
EJDE-2015/11
¯ ε)] = 5c(B + ε[φ])(
b ξ¯ +
[V̂ (θ, ξ,
√
εI)C 4 (ϕT ).
Therefore,
¯ ε) = V̂ (θ, ξ,
¯ ε) − [V̂ (θ, ξ,
¯ ε)]
Ve (θ, ξ,
¯ 4 (ϕT )(ε(φb − [φ]))
b + O(ε3/2 ) = O(ε).
= 5cξC
We can rewrite the equation (3.1) as follows
v̇ = w,
¯ ξ,
¯ ε)v −
ẇ + Av = −εVe (ω̃(ξ)t,
4
X
¯ ξ,
¯ ε)v k+1 ,
εk Vk+1 (ω̃(ξ)t,
(3.6)
k=1
where A = d4 /dx4 + mε , t ∈ R. As it is well known, the equation (3.6) can be
studied as an infinite dimensional Hamiltonian system by taking the phase space
to be the product of the Sobolev spaces H01 ([0, 2π]) × L2 ([0, 2π]) with coordinates
v and w = ∂t v. The Hamiltonian for (3.6) is then
Z 2π
1
1 e
1
¯
¯
v 2 dx
H = hw, wi + hAv, vi + εV (ω̃(ξ)t, ξ, ε)
2
2
2
0
(3.7)
4
X
¯ ¯ Z 2π
k Vk+1 (ω̃(ξ)t, ξ, ε)
k+2
ε
+
v
dx,
k+2
0
k=1
where ∠·, ·i denotes the usual scalar product in L2 ([0, 2π]). We will find the solutions v(t, x) of (3.6) which satisfy
v(t, −x) = v(t, x),
(t, x) ∈ R × T.
It is easy to see that λj = j + mε (j = 0, 1, . . . ) and φj (x) = √1π cos(jx)
(j = 1, 2, . . . ), φ0 (x) = √12π are, respectively, the eigenvalues and eigenfunctions of
Sturm-Liouville problems
4
Ay = λy,
x ∈ T,
y(−x) = y(x),
(3.8)
and the eigenfunctions φj (x)0 s with j ≥ 0 form a complete orthogonal basis of the
subspace consisting of all even functions of L2 (0, 2π).
We introduce coordinates q = (q0 , q1 , q2 , . . . ), p = (p0 , p1 , p2 , . . . ) through the
relations
Xp
X qj (t)
4
p
v(t, x) =
φ
(x),
∂
v(t,
x)
=
λj pj (t)φj (x).
(3.9)
j
t
4
λ
j
j≥0
j≥0
The coordinates are taken from some real Hilbert space:
n
la,s = la,s (R) := q = (q0 , q1 , q2 , . . . ), qi ∈ R, i ≥ 0 such that
o
X
kqk2a,s = |q0 |2 +
|qi |2 i2s e2ai < ∞ .
i≥1
Next we assume that a ≥ 0 and s > 1/2. One rewrites the Hamiltonian (3.7) in
the coordinates (q, p),
H = Λ + G,
(3.10)
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QUASI-PERIODIC SOLUTIONS
7
where
Λ=
¯ ξ,
¯ ε)
1 Xp
Ve (ω̃(ξ)t,
p
λj (p2j + qj2 ) + ε
qj2 ,
2
2 λj
j≥0
G=
4
X
¯
ε
¯
k Vk+1 (ω̃(ξ)t, ξ, ε)
k+2
k=1
2π
Z
0
X q (t)
k+2
j
p
φ
(x)
dx.
j
4
λj
j≥0
The equations of motion are
¯ ε)
p
∂H
Ve (θ, ξ,
∂G
= − λj qj − ε p
,
qj −
∂qj
∂qj
λj
P
with respect to the symplectic structure
dqi ∧ dpi on la,s × la,s .
q˙j =
p
∂H
= λj pj ,
∂pj
p˙j = −
j ≥ 0 (3.11)
Lemma 3.1. Let I be an interval and let
t ∈ I → (q(t), p(t)) ≡ ({qj (t)}j≥0 , {pj (t)}j≥0 )
be a real analytic solution of (3.11) for a > 0. Then
X qj (t)
p
v(t, x) =
φj (x)
4
λj
j≥0
is a classical solution of (3.1) that is real analytic on I × [0, 2π].
One introduces a pair of action-angle variables (J, θ), where J ∈ Rm+1 is canon¯ ∈ Tm+1 . Then (3.6) can be written as a Hamiltonian
ically conjugate to θ = ω̃(ξ)t
system
∂H
∂H
∂H
¯
(3.12)
q˙j =
, p˙j = −
, j ≥ 0, θ̇ = ω̃(ξ),
J˙ = −
∂pj
∂qj
∂θ
with the Hamiltonian
Xp
¯ ε)
Ve (θ, ξ,
¯ Ji + 1
H = hω̃(ξ),
λj (p2j + qj2 ) + ε p
qj2
2
2
λ
j
(3.13)
j≥0
¯ ε) + ε2 G4 (q, θ, ξ,
¯ ε) + ε3 G5 (q, θ, ξ,
¯ ε) + ε4 G6 (q, ϑ),
+ εG3 (q, θ, ξ,
where ϑ = ωt,
¯ ε) =
G3 (q, θ, ξ,
X
¯ ε)qi qj qd
G3i,j,d (θ, ξ,
(3.14)
i,j,d≥0
with
¯ ε) Z
1 V2 (θ, ξ,
φi (x)φj (x)φd (x)dx.
= p
3 4 λi λj λd T
It is not difficult to verify that, from the definition of eigenfunctions,
¯ ε) = 0, unless i ± j ± d = 0 .
G3 (θ, ξ,
¯ ε)
G3i,j,d (θ, ξ,
i,j,d
4
5
Expressions G , G and G6 are similar. We introduce complex coordinate
1
1
zj = √ (qj − ipj ), z̄j = √ (qj + ipj ), j ≥ 0,
2
2
that live in the now complex Hilbert space:
la,s = la,s (C)
n
:= z = (z0 , z1 , z2 , . . . ), zj ∈ C, j ≥ 0such that
(3.15)
8
Q. TUO, J. SI
kzk2a,s = |z0 |2 +
EJDE-2015/11
X
o
|zj |2 j 2s e2aj < ∞ .
j≥1
√
This transformation is symplectic with dq ∧ dp = −1dz ∧ dz̄. Then (3.13) is
changed into
¯ ε) + ε2 G4 (z, θ, ξ,
¯ ε) + ε3 G5 (z, θ, ξ,
¯ ε) + ε4 G6 (z, θ, ε), (3.16)
H = H̃ + εG3 (z, θ, ξ,
where
¯ Ji +
H̃ = hω̃(ξ),
Xp
¯ ε)
εVe (θ, ξ,
2
p
λj zj z̄j +
(zj + z̄j ) ,
4
λ
j
j≥0
+ z̄0 3
¯ ε) = G3 (θ, ξ,
¯ ε) z0√
G3 (z, θ, ξ,
0,0,0
2
X
zj + z̄j zd + z̄d z0 + z̄0 3
¯
√
√
+3
G0,j,d (θ, ξ, ε) √
2
2
2
j,d6=0
X
+ z̄i zj + z̄j zd + z̄d ¯ ε) zi√
√
√
.
+
G3i,j,d (θ, ξ,
2
2
2
i,j,d6=0
(3.17)
(3.18)
Expressions G4 , G5 , G6 are defined similarly to G3 .
4. Reducibility of linear Hamiltonian system
In this section, we are concerned with the reducibility of linear quasi-periodic
Hamiltonian system (3.17). Our result shows that system (3.17) can be reduced to
constant coefficients for any fixed ω ∈ DΛ . Let us rewrite Hamiltonian (3.17) as
H̃ = H0 + εH1 ,
(4.1)
where
¯ Ji +
H0 = hω̃(ξ),
Xp
λj zj z̄j ,
j≥0
H1 =
X Ve (θ, ξ,
¯ ε)
p
(zj + z̄j )2 .
4
λ
j
j≥0
4.1. Reducibility theorem.
Theorem 4.1. Consider the Hamiltonian H̃ given by equation (4.1). Then there
is a 0 < ε∗∗ < ε∗ , 0 < % < 1 and a set J ⊂ Jˆ with meas J ≥ meas Jˆ 1 − O(%)
¯ ∈ Aγ there is a linear symplectic
such that for any 0 < ε < ε∗∗ , ξ¯ ∈ J and α(ξ)
transformation
Σ∞ : Da,s (σ/2, r/2) × J → Da,s (σ, r)
such that the following statements hold:
(i) There is some absolute constant C > 0 such that
|Σ∞ − id |∗a,s+1, Da,s (σ/2,r/2)×J ≤ Cε,
where id is identity mapping.
(ii) The transformation Σ∞ changes Hamiltonian (4.1) into
X
¯ Ji +
H̃ ◦ Σ∞ = hω̃(ξ),
µj zj z̄j ,
j≥0
where
µj =
∞
X
p
¯ ω̃(ξ),
¯ ε),
λj +
εk λ̃j,k (ξ,
k=1
(4.2)
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QUASI-PERIODIC SOLUTIONS
¯ ε)]
e ξ,
¯ ω̃(ξ),
¯ ε) = [V (θ,
p
λ̃j,1 (ξ,
= 0,
2 λj
¯ ω̃(ξ),
¯ ε) = [ζj,k−1,1,1 ], |λ̃j,k (ξ,
¯ ω̃(ξ),
¯ ε)| ≤ C,
λ̃j,k (ξ,
9
k = 2, 3, . . . .
4.2. Regularity of the perturbation term. Let
G3i,j,d = G3|i|,|j|,|d| ,
G4ijdl = G4|i||j||d||l| , G5ijdlm = G5|i||j||d||l||m| ,
G6ijdlmn = G6|i||j||d||l||m||n| .
Noting that the transformation Σ∞ is linear, and from (i) of Theorem 4.1 we get
for j = 0, 1, 2, . . .
∗
¯ ε)zj + εf˜∗ (θ; ξ,
¯ ε)z̄j ,
zj ◦ Σ∞ = zj + εf˜j,∞
(θ; ξ,
∞,j
where
∗
¯ ε)k∗
,
kf˜j,∞
(θ; ξ,
Θ(σ/2)×J
∗
¯ ε)k∗
kf˜∞,j
(θ; ξ,
≤ C.
Θ(σ/2)×J
For convenience we introduce another coordinates (. . . , w−2 , w−1 , w0 , w1 , w2 , . . . ) in
lbs by letting z0 = w0 , z̄0 = w−0 , zj = wj , z̄j = w−j where lbs consists of all bi-infinite
sequence with finite norm
kwk2a,s
2
2
= |w0 | + |w−0 | +
∞
X
|wj |2 |j|2s e2a|j| .
|j|≥1
Hamiltonian (3.17) is changed into
¯ Ji +
b := H̃ ◦ Σ∞ = hω̃(ξ),
H
X
µj wj w−j ,
(4.3)
j≥0
¯ ε)w0 + S12 (θ, ξ,
¯ ε)w−0 ,
(z0 + z¯0 ) ◦ Σ∞ = S11 (θ, ξ,
where
¯ ε) := 1 + εf˜1 (θ; ξ,
¯ ε),
S11 (θ, ξ,
0,∞
¯ ε) := 1 + εf˜2 (θ; ξ,
¯ ε)
S12 (θ, ξ,
0,∞
with
1
¯ ε)k∗
kf˜0,∞
(θ; ξ,
,
Θ(σ/2)×J
2
¯ ε)k∗
kf˜0,∞
(θ; ξ,
≤ C.
Θ(σ/2)×J
This implies the Hamiltonian (3.16) is changed by the transformation Σ∞ into
b + εG̃3 + ε2 G̃4 + ε3 G̃5 + ε4 G̃6 .
H=H
3
(4.4)
6
Next we consider the regularity of the gradient of G̃ , . . . , G̃ .
a,s
Lemma 4.2. For a ≥ 0 and s > 1/2, the space
P l is a Hilbert algebra with respect
to convolution of the sequences, (q ∗ p)j := k qj−k pk , and
kq ∗ pka,s ≤ Ckqka,s kpka,s
with a constant C depending only on s.
The proof for the above lemma is similar to that of [23, Lemma A]. Using the
above lemma, we can prove the following Lemma.
Lemma 4.3. For a ≥ 0 and s > 1, the gradient G̃3w , G̃4w , G̃5w and G̃6w are real
analytic for real argument as a map from some neighborhood of origin in la,s into
la,s+1/2 , with
kG̃3w ka,s+1/2 ≤ Ckwk2a,s ,
kG̃4w ka,s+1/2 ≤ Ckwk3a,s , . . . , |G̃6w ka,s+1/2 ≤ Ckwk5a,s
10
Q. TUO, J. SI
EJDE-2015/11
¯ ∈ Θ(σ/2) × J,
ˆ where C is a constant large enough as ε small
uniformly for (θ, ξ)
3
enough. The Hamiltonian G̃ till G̃5 depend on the “time” θ = (θ1 , . . . , θm , θm+1 ) =
¯
(ω1 t, . . . , ωm t, α(ξ)t).
5. The Birkhoff normal form
In this section, we transform the hamiltonian (4.4) into some partial Birkhoff
normal form of order six so that it appears, in a sufficiently small neighbourhood
of the origin, as a small perturbation of some nonlinear integrable system. To this
end we have to kill the perturbation G̃3 , G̃4 , G̃5 and the non-resonant part of the
perturbation G̃6 by Birkhoff normal form.
By XF1 3 denote the time-1 map of the vector field of the Hamiltonian εF3 , Then
letting
b F3 } = 0,
G̃3 + {H,
b F3 } = 0. Therefore,
we have found a quasi-periodic function F3 such that G̃3 + {H,
we get the new Hamiltonian
b + ε2 G 4 + ε3 G 5 + ε4 G 6 + ε5 R1 ,
H=H
4
(5.1)
5
Repeating this process, we eliminate all terms in G and G . Hamiltonian (5.1) are
changed into
b + ε4 G6 + ε5 R11 + ε6 R22 + ε7 R33 + ε8 R44 + ε9 R55 + ε10 R66 ,
H=H
(5.2)
and we can write
G6 =
X
β
s
G6rsαβ w0r w−0
wjα w−j
.
(5.3)
|α|+|β|+r+s=6
Let Ln = {(i, j, d, l, m, n) ∈ Z6 : 0 6= min(|i|, |j|, |d|, |l|, |m|, |n|) ≤ n}, and Nn ⊂ Ln
be the subset of all (i, j, d, l, m, n) ≡ (i, −i, j, −j, l, −l). That is, they are of the
form (i, −i, j, −j, l, −l) or some permutation of it.
We define the indices set ∆∗ , ∗ = 0, 1, 2, 3. For each ∗ = 0, 1, 2, ∆∗ is the set of
indices {i, j, d, l, m, n} which have exactly ∗ components not in Ln , ∆3 is the set
which has at least three components not in Ln . then we split (5.3) into three parts:
6
b6 + G
e 6,
G6 = G + G
6
where G is the normal form part of G6 , with (i, . . . , n) ∈ (∆0 ∪ ∆1 ∪ ∆2 ) ∩ Nn :
X
6
G =
G6iijjll |wi |2 |wj |2 |wl |2 ,
i∈Nn , j,l>1
b 6 is the normal form part of G6 , with (i, . . . , n) ∈ (∆0 ∪ ∆1 ∪ ∆2 )\Nn :
G
X
b6 =
G
G6ijdlmn wi wj wd wl wm wn ,
(i,...,n)∈(∆0 ∪∆1 ∪∆2 )\Nn
e 6 is the normal form part of G6 , with (i, . . . , n) ∈ 43 :
G
X
e6 =
G
G6ijdlmn wi wj wd wl wm wn .
(i,...,n)∈43
Using the same methods as in Section 5.1, there is a hamiltonian F6 which has the
b 6 by a symplectic transformation X 1 ,
same form as that of G6 , and will eliminate G
F6
EJDE-2015/11
QUASI-PERIODIC SOLUTIONS
11
which is the time-1-map of the flow of a hamiltonian vector XF6 with hamiltonian
ε4 F6 , let
X
¯ ε)]w3 w3 + w2 w2
¯ ε)wj w−j
b F6 } + G6 = [χ33 (θ, ξ,
{H,
[G622jj ](θ, ξ,
0 −0
0 −0
j≥1
+ w0 w−0
¯ ε)wi w−i wj w−j + G6 + G
e 6.
[G600iijj ](θ, ξ,
X
i∈Nn ,j≥1
By a direct calculation, we obtain the following lemma.
Lemma 5.1. For each finite n ≥ 1, there exists a real analytic, symplectic change
of coordinates XF1 6 in some neighborhood of the origin on the complex Hilbert space
la,s such that the hamiltonian (5.2) is changed into
X
X
b + c0 ε4 z03 z̄03 + ε4 z02 z̄02
H ◦ XF16 = H
cj zj z̄j + ε4 z0 z̄0
[G600iijj ]zi z̄i zj z̄j + ε4 K,
j≥1
i,j≥1
where
6
6
9
Z
K = G + Ĝ + εR11 + εR22 + · · · + ε
0
c0 =
cj =
[φ]
1
{R66 , F6 } ◦ XFs 6 ds,
1
24π 2 [V̂ ]3/2
ε−3/4 (1 + O(ε 4 )),
[φ]
24π 2 [V̂
p ε−1/2 (1 + O(ε1/2 )),
] λj
and [G600iijj ] denotes the 0-Fourier coefficient of G600iijj with

¯ ε)), i 6= j,
 6! [G600iijj ] = √15[φ] (1 + O(ε)) + $ij (ξ,
48
6
π 2 λ0 λi λj
[G00iijj ] =
15[φ]
¯
 6! [G6
√
i = j,
00iiii ] = 4π 2 λ λ (1 + O(ε)) + $ij (ξ, ε)),
48×2
i
(5.4)
0
and
G6iijjll =
48π 2
[φ]
p
(4 + 2δij + 2δjl + 2δjk + 2δi+j,l + 2δi+l,j + 2δl+j,i ). (5.5)
λi λj λl
Here
(
1,
δij =
0,
i = j,
i 6= j,
¯ ε) depends smoothly on ξ¯ and ε and there is an absolute constant C such that
$ij (ξ,
¯ ε)k∗ ≤ Cε for ε small enough, while Ĝ is only dependent on the coordinates
k$ij (ξ,
J
ẑ and we have
|Ĝ| = O(kẑk6a,s ), |K| = O(kzk7a,s )
uniformly for | Im θ| < σ/5, ξ¯ ∈ J , ẑ = (zj )j∈N\N .
d
We introduce the action-angle variable by setting
(p
Ij e−iθ̂j , j ∈ Nd ∪ {0},
zj =
zj = zj ,
j∈
/ Nd ∪ {0}.
By the symplectic change (5.6), the normal form becomes
X
X
b + c0 ε4 z 3 z̄ 3 + ε4 z 2 z̄ 2
H
cj zj z̄j + ε4 z0 z̄0
cj |zi |2 |zj |2
0 0
0 0
j∈N
i∈Nn ,j≥1
(5.6)
12
Q. TUO, J. SI
¯ Ji + µ0 I0 + c0 ε4 I 3 +
= hω̃(ξ),
0
X
EJDE-2015/11
(µj + ε4 I02 cj )Ij +
j∈Nd
X
(µj + ε4 I02 cj )zj z̄j
j ∈N
/ d
3.5
ε
+ I0 q
1
( hAI, Ii + hBI, Ẑi)
2
[Vb ]
b = (|zd+1 |2 , |zd+2 |2 , . . . ),
with I = (I1 , . . . , Id ), A = (aij )i,j∈Nd , B = (aij )j∈N,i∈Nd , Z
and

¯ ε)), i 6= j,
 215[φ]
√
(1 + O(ε)) + $ij (ξ,
aij = π λi λj
 15[φ] (1 + O(ε)) + $ (ξ,
¯ ε)),
i = j.
ij
4π 2 λi
Now let us introduce the parameter vector ξ˜ = (ξ˜j )j∈Nd ∪{0} and the new action
variable and ρ̃ = (ρ̃j )j∈Nd ∪{0} as follows
Ij = εξ˜j + ρ̃j ,
ξ˜j ∈ [1, 2],
|ρ̃j |hε4 ,
j = {0} ∪ Nd .
Clearly, dθ̂j ∧ dIj = dθ̂j ∧ dρ̃j . So the transformation is symplectic. Then the
normal form is changed into
X
X
¯ Ji + µ0 + 3c0 ε6 ξ˜2 + 2ε6 ξ˜0
hω̃(ξ),
cj ξ˜j ρ̃0 +
(µj + ε6 ξ˜02 cj )ρ̃j
0
j∈Nd
+
(3c0 ε5 ξ˜0
+
X
+ε
X
5
j∈Nd
X
cj ξ˜j )ρ̃20 + 2ε5 ξ˜0 ρ̃0
j∈Nd
j∈Nd
+
[G600iiii ]ρ̃i ξ˜j
+
X
ε6 ξ˜0
[G600iiii ]ξ˜i ρ̃j
[G600iiii ]ξ˜j ρ̃i + ε5 ρ̃0
X
+ ε6 ξ˜0
i,j∈Nd
X
X
[G600iiii ]ξ˜i |zj |2
i∈Nd ,j ∈N
/ d
X
[G600iiii ]ξ˜i ρ̃j + ε5 ρ̃0
i,j∈Nd
i∈Nd ,j ∈N
/ d
+ ε (ξ˜0 + ρ̃0 )2
4
[G600iiii ]ρ̃i ρ̃j
i,j∈Nd
X
i,j∈Nd
+ ε5 ρ̃0
X
(µj + ε (εξ˜0 + ρ̃0 )2 cj )zj z̄j + ε5 ξ˜0
4
j ∈N
/ d
ε6 ξ˜0
cj ρ̃j + c0 ε4 ρ̃30 +
X
[G600iiii ]ρ̃i ρ̃j
i,j∈Nd
[G600iiii ]ρ̃i |zj |2 .
i∈Nd ,j ∈N
/ d
Hence, the total Hamiltonian is
¯ Ji + ω̌0 ρ̃0 +
H = hω̃(ξ),
X
X
ω̌j ρ̃j +
j∈Nd
X
+ ε6 ξ˜0
[G600iijj ]ρ̃i ξ˜j + ε6 ξ˜0
i,j∈Nd
X
[G600iijj ]ξ˜i ρ̃j
i,j∈Nd
X
+ ε6 ξ˜0
λ̌j zj z̄j
j∈N
[G600iijj ]ξ˜i |zj |2 + P,
i∈Nd ,j ∈N
/ d
where
ω̌0 = µ0 + 3c0 ε6 ξ˜02 + 2ε6 ξ˜0
X
cj ξ˜j ,
j∈Nd
ω̌j = µj +
ε6 ξ˜02 cj
+
ε6 ξ˜0
X
[G600iiii ]ξ˜i ,
j ∈ Nd
i,j∈Nd
λ̌j = µj + ε6 ξ˜02 cj + ε6 ξ˜0
X
i∈Nd ,
[G600iiii ]ξ˜i ,
j∈
/ Nd ,
(5.7)
EJDE-2015/11
QUASI-PERIODIC SOLUTIONS
X
P = (3c0 ε5 ξ˜0 + ε5
cj ξ˜j )ρ̃20 + 2ε5 ξ˜0 ρ̃0
j∈Nd
5
+ ε ρ̃0
X
+ ε (ξ˜0 + ρ̃0 )
cj ρ̃j + c0 ε4 ρ̃30
j∈Nd
[G600iiii ]ξ˜j ρ̃i
X
5
+ ε ρ̃0
i,j∈Nd
4
X
[G600iiii ]ξ˜i ρ̃j + ε5 ρ̃0
i,j∈Nd
X
2
[G600iiii ]ρ̃i |zj |2
+ε
+ε
5
X
G6 ξ˜l ρ̃i ρ̃j
+ε
X
G6 ξ˜i ρ̃l ρ̃j + ε5
G6 ξ˜i ρ̃j zl z l
+ε
X
+ε
X
G ρ̃i ρ̃j zl z l + ε
4
6
X
G6 ξ˜j ρ̃i ρ̃l
G6 ξ˜j ρ̃i zl z l
X
5
ξ˜i zj z j zl z l
i∈Nd ,j,l∈N
/ d
X
ρ̃i zj z j zl z l + ε4
i∈Nd ,j,l∈N
/ d
4
G ρ̃i ρ̃j ρ̃l
i,j∈Nd ,l∈N
/ d
6
i,j∈Nd ,l∈N
/ d
+ ε4
6
i,j,l∈Nd
X
5
i,j∈Nd ,l∈N
/ d
4
X
i,j,l∈Nd
X
[G600iiii ]ρ̃i ρ̃j
i,j,l∈Nd
+ ε5
i,j,l∈N
X
i,j∈Nd
4
i∈Nd ,j ∈N
/ d
5
13
G6ijdlmn zi zj zd zl zm zn + ε5 K
(i,...,n)∈43
5
5
= ε Q + ε Ĝ + ε K̂ + ε K,
with
Q = O(|ρ̃|3 ) + O(|ρ̃|kẐk2 ),
Ĝ6 =
X
G6ijdlmn zi zj zd zl zm zn ,
(i,...,n)∈43
K̂ = O(|ρ̃|2 ) + O(|ρ̃|kẐk),
K = R11 + R22 + · · · + ε9
Z
0
1
{R66 , F6 } ◦ XFs 6 ds.
Next, we give the estimates of the perturbed term P . To this end we need some
notation which is taken from [23]. Let la,s is now the Hilbert space consisting of
those vectors Z with kZka,s < ∞. with
X
kZk2a,s =
|zj |2 |j|2s e2aj < ∞, a, s > 0.
j ∈N
/ d
Let x = (θ, θ̂0 ) ⊕ θ̂, with θ̂ = (θ̂j )j∈Nd , y = (J, ρ̃0 ) ⊕ ρ̃, ρ̃ = (ρ̃j )j∈Nd , Z = (zj )j ∈N
/ d,
and let us introduce the phase space
b m+d+2 × Cm+d+2 × la,s × la,s 3 (x, y, Z, Z̄),
P a,s = T
b m+d+2 is the complexiation of the usual (m + d + 2)-torus Tm+d+2 . Set
where T
D(s, r) := {(x, y, Z, Z̄) ∈ P a,s : | Im x| < s, |y| < r2 , kZka,s + kZ̄ka,s < r}.
We define the weighted phase norms
|W |r = |W |s̄,r = |x| +
1
1
1
|y| + kZka,s̄ + kZ̄ka,s̄
r2
r
r
for W = (x, y, Z, Z̄) ∈ P a,s̄ with s̄ = s + 1. Denote by Σ the parameter set
J × [1, 2]m+d+1 . For a map U : D(s, r) × Σ → P a,s̄ , define its Lipschitz semi-norm
|U |L
r:
|∆ξ̂ξ U |r
|U |L
,
r = sup ˆ
ξ̂6=ξ |ξ − ξ|
14
Q. TUO, J. SI
EJDE-2015/11
ˆ
where ∆ξ̂ξ U = U (·, ξ)−U
(·, ξ), and where the supremum is taken over Σ. Denote by
XP the vector field corresponding the Hamiltonian P with respect to the symplectic
structure dx ∧ dy + idZ ∧ dZ̄, namely,
XP = (∂y P, −∂x P, ∇Z̄ P, −∇Z P ).
Lemma 5.2. The Perturbation P (x, y, Z, Z̄; ζ) is real analytic for real argument
(x, y, Z, Z̄) ∈ D(s, r) for given s, r > 0, and Lipschitz in the parameters ξ ∈ Σ, and
for each ξ ∈ Σ its gradients with respect to Z, Z̄ satisfy
∂Z P,
∂Z̄ P ∈ A(la,s , la,s+1/2 ),
where A(la,s , la,s+1/2 ) denotes the class of all maps from some neighborhood of the
origin in la,s into la,s+1/2 , which is real analytic in the real and imaginary parts of
the complex coordinate Z. In addition, for the perturbed term P we have the two
estimates
sup |∂ξ XP |r ≤ Cε4 ,
sup |XP |r ≤ Cε4 ,
D(s,r)×Σ
D(s,r)×Σ
where s = σ/5 and r = ε.
Proof. For j ∈ Nd , from (5.6) it follows that |∂θ̂j wj | ≤ Cε1/2 and |∂ρ̃j wj | ≤ Cε−1/2
where wj = zj or wj = z̄j . From (5.6) and kZka,s ≤ r = ε, we obtain kzka,s ≤ Cε1/2
6
where z = (z0 , z) ⊕ Z with z = (zj ∈ C : j ∈ Nd ). In view of |G | = O(ε8 ), |Ĝ6 | =
O(ε6 ) and K = O(ε17/2 ), it follows that |P | = O(ε10 ) on D(s, 2r). Using Cauchy
estimates for ∂x P , ∂y P , ∂Z̄ P and ∂Z P , we obtain |∂x P | = O(ε10 ), |∂y P | = O(ε8 ),
|∂Z̄ P | = O(ε9 ), |∂Z P | = O(ε9 ) on D(s0 , r). Hence, we have supD(s,r)×Σ |XP |r ≤
Cε4 . By a direct computation with respect to ξ, we also have supD(s,r)×Σ |∂ξ XP |r ≤
Cε4 .
6. Proof of main theorem
To apply the infinite-dimensional KAM theorem which was first proved by Kuksin
[18, 19] and Pöschel [23] to our problem, we need to introduce a new parameter ω̄
below.
¯ ∈ Aγ . Hence, for fixed ω− = (ω 1 , ω 2 , . . . , ω m ) ∈
For any ξ¯ ∈ J , we have α(ξ)
−
−
−
m+1 ¯
DΛ and ω− (ξ) ∈ Aγ arbitrarily. For
n
¯ := ω̃(ξ)
¯ ∈ Ω̄
¯ = (ω1 , . . . , ωm , α(ξ))
¯ ∈ DΛ × Aγ : |ωi − ω i | ≤ ε,
ω̃(ξ)
−
o
¯ − ω m+1 (ξ)|
¯ ≤ε ,
|α(ξ)
−
we can introduce new parameter ω̄ = (ω̄1 , ω̄2 , . . . , ω̄m , ω̄m+1 ) by the following
j
ωj = ω−
+ ε6 ω̄j , ω̄j ∈ [0, 1], j = 1, 2, . . . , m,
¯ = ω m+1 (ξ)
¯ + ε6 ω̄m+1 , ω̄m+1 ∈ [0, 1].
α(ξ)
−
Hence, the Hamiltonian (5.7) becomes
b
H = hb
ω (ξ), ŷi + hΩ(ξ),
Ẑi + P,
¯ ⊕ ω̌0 ⊕ ω̆ with
where ω
b (ξ) = ω̃(ξ)
ε5.5 ξ˜0 ˜
ω̆ = α̃ + p
Aξ,
[V̄ ]
ξ˜0
˜
b
Ω(ξ)
= β̃ + ε5.5 p B ξ,
[V̄ ]
(6.1)
EJDE-2015/11
QUASI-PERIODIC SOLUTIONS
˜
ξ = ω̄ ⊕ ξ˜0 ⊕ ξ,
ŷ = J ⊕ ρ̃0 ⊕ ρ̃,
15
ξ˜ := (ξ˜j )j∈Nd ,
α̃ = (ω̌1 , . . . , ω̌d ),
β̃ = (λ̌d+1 , λ̌d+2 , . . . ).
Lemma 6.1. Let Π = [1, 2]m+d+2 . Then we have XP ∈ A(la,s , la,s+1/2 ) and
|XP |r ≤ Cε4 ,
sup
|∂ζ XP |r ≤ Cε4 .
sup
D(s,r)×Π
D(s,r)×Π
The proof of the above lemma is the same as one of the lemma 5.2. In the
following, we verify the assumptions A, B and C in [23], for the above Hamiltonian
(6.1). Recalling (5.4), we have
A = (aij )
 b
=
¯
√
λi1 + $11 (ξ, ε)
√ a
¯ ε)
 λ λ + $21 (ξ,
 i2 i1

√ a
λid λi1
...
¯ ε)
+ $d1 (ξ,
a
λi(d+1) λi1
√ a
 λ
 i(d+2) λi1
√
B=
a
λi1 λi2
b
λi2 +
√ a
λid λi2
¯ ε)
√ a
+ $1d (ξ,
λi1 λid
¯ ε) 
√ a

+ $4 (ξ,
λi2 λid

¯ ε) . . .
+ $12 (ξ,
¯ ε)
$22 (ξ,
...
...
...
¯ ε) . . .
+ $n2 (ξ,
¯ ε) . . .
+ $d+1,1 (ξ,
¯ ε) . . .
+ $d+2,1 (ξ,
..
.
b
λid
d×d
¯ ε)
+ $d+1,d (ξ,
¯ ε)
+ $d+2,d (ξ,


..
.
a
λi(d+1) λid
√ a
λi(d+2) λid
√
..
.
,

...
¯ ε)
+ $dd (ξ,
,
∞×d
where
a=
15[φ]
,
π2
b
i2 ×i2
 1a 1
 i22 ×i21
a
i21 ×i22
b
i22 ×i22
a
i2d ×i21
a
i2d ×i22

lim A = 
 ...
...
ε→0

a
i2d+1 ×i21
 a
2
2
lim B = 
 id+2 ×i1
..
.
ε→0
b=
15[φ]
.
4π 2
a 
i21 ×i2d
a 
i22 ×i2d 
...
...
...
...
:= D,

... 
b
i2d ×i2d
a
...
(6.2)
i2d+1 ×i2d
d×d


a
i2d+2 ×i2d 
...
..
.
..
.
e
:= D.

∞×d
Setting û = (i21 , i22 , . . . , i2d ) and v̂ = (i2d+1 , i2d+2 , . . . ), and defining the matrices
Ē := diag[û],
F̄ := diag[v̂],
e as
we can rewrite D and D
D = Ē −1 AĒ −1 ,
e = F̄ −1 B Ē −1 ,
D
where

b
 a
A=
. . .
a
a
b
...
a
...
...
...
...

a
a

,
. . .
b d×d

b
b
B=
..
.
...
...
..
.

b
b
.

..
. ∞×d
16
Q. TUO, J. SI
EJDE-2015/11
We know that det D 6= 0 since det A = (b − a)d−1 (b + (d − 1)a) 6= 0 (a, b have the
same sign). Therefore, we get det A 6= 0 provided that 0 < ε 1. Moreover, by
the definition of ω
b , we get that


Im+1
0
0
P
∂ω
b
∂(e
ω , ω̆0 , ω̆)
6c0 + j∈Nd cj ξj 12Y , for ξ ∈ Π,
=
= ε6 ξ˜0  0
e
∂ξ
∂(ω, ξe0 , ξ)
0
12ξ˜0 YT + Aξ
A
where Im+1Pdenotes the (m + 1) × (m + 1) unit matrix, Y = (c1 , c2 , . . . , cd ). Let
c00 = 6c0 + j∈Nd cj ξj . In view of c0 = O(ε−3/4 ), cj = O(ε−1/2 ) and
c00
12Y
1
0
1 −YA−1
0
Id
12YT + Aξ
A
−A−1 YT Id
0
c − YA−1 YT − Yξ 0
= 0
,
Aξ
A
we get
c00 − YA−1 YT − Yξ 0
det
6= 0
Aξ
A
provided that 0 < ε 1. Therefore, the real map ξ 7→ ω
b (ξ) is a lipeomorphism
between Π and its imagine.
For any k ∈ Zm+2+d , we write
k = (k1 , k2 , k3 ),
k1 ∈ Zm+1 , k2 ∈ Z, k3 ∈ Zd .
Let
b
Y(ξ) = hk, ω
b (ξ)i + hl, Ω(ξ)i
˜
˜
¯ + k2 ω̌0 + hk3 , α̃i + hk3 , ε5.5 qξ0 Aξi
= hk1 , ω̃(ξ)i
b
[V ]
ξ˜0
˜
+ hl, β̃ + ε5.5 q B ξi,
[Vb ]
∆ := {ξ ∈ Π : Y(ξ) = 0}.
We need to prove that meas ∆ = 0. We discuss the following two cases.
Case 1. Let k1 = (k 1 , k 2 , . . . , k m , k m+1 ) 6= 0 and write
¯ =
hk1 , ω̃(ξ)i
m
X
¯
k i ωi + k m+1 α(ξ),
i=1
then there exists some 1 ≤ i0 ≤ m such that k i0 6= 0. Observe that ω̌0 , α̃ and β̃ do
not involve the parameter ω̄. Then
∂Y(ξ)
= k i0 ε6 6= 0,
∂ ω̄i0
0 < ε 1,
which implies meas ∆ = 0.
Case 2. Let k2 6= 0. Then
X
∂Y(ξ)
= 6c0 ε6 + 2ε6
cj ξ˜j 6= 0,
∂ ξ˜0
j∈Nd
which implies meas ∆ = 0.
EJDE-2015/11
QUASI-PERIODIC SOLUTIONS
17
Case 3. Let k1 = k2 = 0, then
ε5.5 ˜
˜
Y(ξ) = hk1 , ωi + k2 ω̌0 + hk3 , α̃i + hk3 , q Aξi
+ hl, β̃ + ε3 B ξi
ˆ[V ]
ε5.5 ˜
ε5.5 ˜
= hk3 , α̃i + hk3 , q Aξi
+ hl, β̃ + q B ξi
ˆ[V ]
ˆ[V ]
ε5.5
˜
= hk3 , α̃i + hl, β̃i + q hAk3 + B T l, ξi,
ˆ[V ]
where B T is the transpose of B. (Note that A is symmetric.) We claim that either
hk3 , α̃i + hl, β̃i =
6 0 or Ak3 + B T l 6= 0.
Since
e T l,
lim (Ak3 + B T l) = Dk3 + D
ε→0
and
lim (hk3 , α̃i + hl, β̃i) = hk3 , α̂i + hl, β̂i,
ε→0
with α̂ = (i21 , i22 , . . . , i2d ) and β̂ = (i2d+1 , i2d+2 , . . . ), it suffices to show that hk3 , α̂i +
e T l 6= 0. The result is proved in in[24, Lemma 6]. Hence, we
hl, β̂i 6= 0 or Dk3 + D
get that hk3 , α̃i + hl, β̃i =
6 0 or Ak3 + B T l 6= 0 as 0 < ε 1. Moreover, it is easy
b
to that hl, Ω(ξ)i
=
6 0 as 0 < ε 1, with 1 ≤ |l| ≤ 2 and ξ ∈ Π. This completes the
verification of Assumption A.
Noticing that
q
ε[V̂ ]
+ O(j −4 ),
λj = j 4 + ε[V̂ ] = j 2 +
2j 2
∞
X
p
¯ ω̃(ξ),
¯ ε),
εk λ̃j,k (ξ,
µj = λj +
k=2
b j = j ς + . . . with ς = 2 and Ω
b j − j 2 is a Lipschitz map from Π to
we have that Ω
−δ
b with δ = −2, ς = 2, and
l∞ with δ = −2. Thus, Assumption B is fulfilled for Ω
Ω = β̃.
Assumption C can be verified easily using Lemma 5.2, letting p̄ = s + 1/2, p = s.
Now let us verify the smallness condition in [23]. By letting α = ε8−ι with
0 < ι < 8 fixed and Lemma 6.1, we have
α
sup |XP |r + sup
|XP |L
r ≤ γα,
D(s,r)×Π
D(s,r)×Π M
if 0 < ε < ε∗∗ with a constant ε∗∗ = ε∗∗ (γ, C). This implies the smallness condition
is satisfied. Next, Let us check the conditions of [23, Theorem D] for the Hamiltonian (6.1). First of all, we remark that ω
b (ξ) is affine function of the parameter
ξ. and we can choose µ̃ = 1 in [23, Theorem D].
Let us run the infinite-dimensional KAM theorem for Hamiltonian (6.1). Then
there is a subset Πα ⊂ Π with
meas(Π \ Πα ) ≤ ĉLm+d+2 M m+d+2 (diam Π)m+d+2 αµ̃ ≤ Cε−1 αµ̃ < ε6−ι ,
and a Lipschitz continuous family of torus embedding Φ : Tm+d+2 ×Πα → P a,s+1/2 ,
b
and a Lipschitz continuous map ω
b : Πα → Rm+d+2 , such that for each ξ ∈ Πα
18
Q. TUO, J. SI
EJDE-2015/11
the map Φ restricted to Tm+d+2 × {ξ} is a real analytic embedding of an elliptic
b
rotational torus with frequencies ω
b (ξ) for the Hamiltonian H at ξ. Moreover,
b
|ω
b (ξ) − ω
b (ξ)| < cε. We return from the parameter set Π to
¯ × [0, 1]d+1 .
Π∗ (ω , ω m+1 ) = Ω̄
−
−
Let
m+1
),
Π∗ = ∪(ω− ,ωm+1 )∈DΛ ×Aγ Π∗ (ω− , ω−
−
m+1
m+1 ∗
∗
where ω− , ω−
are chosen such that Π∗ (ω−
, ω−
)∩
m+1 ∗
m+1 ∗∗
∗
∗∗
(ω−
, ω−
) 6= (ω−
, ω−
). Hence, we get a subset Π∗α
∗∗
m+1
∗∗
Π∗ (ω−
, ω−
) = ∅ if
∗
⊂ Π such that
Σα = Π∗α ⊂ DΛ × Aγ × [0, 1]d+1 ⊂ Σ
with
meas(Σ \ Σε ) ≤ ε.
Therefore, for the new parameter set, we have that there are a Lipschitz continuous
family of torus embedding Φ : Tm+d+2 × Σε → P a,s+1 , and a Lipschitz continuous
b
map ω
b : Σε → Rm+d+2 , such that for each ξ ∈ Σε the map Φ restricted to Tm+d+2 ×
{ξ} is a real analytic embedding of an elliptic rotational torus with frequencies
b
¯ (ω̂j )j∈N ∪{0} ) for the Hamiltonian H at ξ. Also
ω
b (ξ) = (ω̃(ξ),
d
α
3−(3−ι)
|Φ − Φ0 |L
= cει ,
(6.3)
|Φ − Φ0 |r +
r ≤ cε
M
α b
b
|ω
b − ω 0 |r +
|ω
b (ξ) − ω 0 (ξ)|L ≤ cε3 ,
(6.4)
M
¯ ξ˜0 , ξ˜1 , . . . , ξ˜d ). Therefore, all motions starting
where ω 0 (ξ) = ω
b (ξ) and ξ = (ω, α(ξ),
m+d+2
b
from the torus Φ(T
× Σε ) are quasi-periodic with frequencies ω
b (ξ). By (6.3)
and (6.4), those motions can written as follows:
ρ̃0 (t) = O(ε3 ),
ρ̃j (t) = O(ε3 ),
θ̂0 (t) = ω̂0 t + O(ει ),
θ̂j (t) = ω̂j t + O(ει ),
kZ(t)ka,s+1 = O(ε),
j ∈ Nd ,
¯
θ(t) = ω̃(ξ)t,
where Z = (zj )j ∈N
/ d and we have chosen the initial phase θ̂j (0) = 0. Returning the
original equation (1.3), we may get the solution described in Theorem 1.1.
Acknowledgments. This work was partially supported by the National Natural
Science Foundation of China (Grant Nos. 10871117, 11171185) and SDNSF (Grant
No. ZR2010AM013)
References
[1] D. Bambusi; On Lyapunov center theorem for some nonlinear PDE’s: A simple proof, Ann.
Scula Norm. Sup. Pisa CL. Sci., 29(2000), 823-837.
[2] D. Bambusi, S. Graffi; Time quasi-periodic unbounded perturbations of Schrödinger operators
and KAM methods. Comm. Math. Phys., 219 (2001), 465–480.
[3] D. Bambusi, S. Paleari; Families of periodic orbits for some PDE’s in higher dimensions,
Commun. Pure Appl. Anal., 29(2002), 269-279.
[4] M. Berti, P. Bolle, M. Procesi; An abstract Nash-Moser theorem with parameters and applications to PDES, Ann. I. H. Poincaré, 27(2010), 377-399.
[5] Yu. N. Bibikov; Stability of zero solutions of essentially nonlinear one-degree-of freedom
hamiltonian and reversible systems, Differential Equations, 38(2002), 609-614.
EJDE-2015/11
QUASI-PERIODIC SOLUTIONS
19
[6] J. Bourgain; Construction of quasi-periodic solutions for Hamiltonian perturbation of linear
equations and application to nonlinear PDE, International Mathematics Research Notices,
11(1994), 475-497.
[7] J. Bourgain; Green’s function estimates for lattice Schrödinger operators and applications,
Ann. of math.Stud., vol 158, Princeton University Press, Princeton, 2005.
[8] P. Drabek, D. Lupo; On generalized periodic solutions of some nonlinear telegraph and beam
equations, Czech. Math. J., 36(1986), 434-449.
[9] A. Eden, A. J. Milani; Exponential attractors for exyensible beam equations, Nonlinearity,
6(1993), 457-479.
[10] H. L. Eliasson, S. B. Kuksin; On reducibility of Schrödinger equations with quasiperiodic in
time potentials. Comm. Math. Phys., 286(2009),125-135.
[11] M. Feckan; Free vibrations of beams on bearings with nonlinear elastic responses, J. Differential Equations, 154(1999), 55-72.
[12] J. Geng, J. You; A KAM tori of Hamiltonian perturbations of 1D linear beam equations, J.
Math. Anal. Appl., 277(2003), 104-121.
[13] J. Geng, J. You; A KAM theorem for Hamiltonian partial differential equations in higher
dimensional spaces, Commun. Math. Phys., 262(2006), 343-372.
[14] J. Geng, J. You; KAM tori for higher dimensional beam equations with contant potentials,
Nonlinearity, 19(2006), 2405-2423.
[15] J. Geng, J. You; A KAM theorem for one dimensional Schrödinger equation with periodic
boundary conditions, J. Differential Equations, 209(2005), 1-56.
[16] B. Grébert, L. Thomann; KAM for the quantum harmonic oscillator, Comm. Math. Phys.,
307 (2011), 383-427.
[17] H. Hanßmann, J. G. Si; Quasi-periodic solutions and stability of the equilibrium for quasiperiodically forced planar reversible and Hamiltonian systems under the Bruno condition,
Nonlinearity, 23(2010),555-577.
[18] S. B. Kuksin; Nearly integrable infinite-dimensional Hamiltonian systems, Lecture Notes in
Math. Vol. 1556, Springer, New York, 1993.
[19] S. B. Kuksin; Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum, Funct. Anal. Appl., 21(1987), 192-205.
[20] C. Lee; Periodic solutions of beam equations with symmetry, Nonlinear Anal., 42(2000),
631-650.
[21] Z. Liang, J. Geng; Quasi-periodic solutions for 1D resonant beam equation, Commu. Pure
Appl. Anal., 5(2006), 839-853.
[22] J. Liu, X. Yuan; Spectrum for quantum Duffing oscillator and small-divisor equation with
large variable coefficent, Commun. Pure Appl. Math., 63(2010), 1145–1172.
[23] J. Pöschel; A KAM-theorem for some nonlinear PDEs, Ann. Sc. Norm. Super. Pisa Cl. Sci.
IV Ser.,(15)23(1996), 119-148.
[24] J. Pöschel; A KAM-theorem for nonlinear wave equation, Comment.Math.Helv.,71(1996),
269-296.
[25] J. G. Si; Quasi-periodic solutions of a non-autonomous wave equations with quasi-periodic
forcing. J. Differential Equations, 252(2012), 5274-5360.
[26] Y. Wang; Quasi-periodic solutions of a quasi-periodically forced nonlinear beam equation,
Commun Nonlinear Sci Numer Simulat., 17(2012), 2682-2700.
[27] H. K. Wang, G. Chen; Asymptotic locations of eigenfrequencies of Euler-Bernoulli beam
with nonhomogeneous structural and viscous damping coefficients, SIAM J. control. Optim.,
29(1991), 347-367.
[28] Y. Wang, J. G. Si; A result on quasi-periodic solutions of a nonlinear beam equation with
a quasi-periodic forcing term, Zeitschrift fur Angewandte Mathematik und Physik (ZAMP),
(1)63(2012), 189-190.
[29] X. Yuan; Quasi-periodic solutions of completely resonant nonlinear wave equations, J. Differential Equations, 230(2005), 213-274.
[30] M. Zhang, J. G. Si; Quasi-periodic solutions of nonlinear wave equations with quasi-periodic
forcing, Physica D, 238(2009), 2185-2215.
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Q. TUO, J. SI
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Qiuju Tuo
School of Mathematics, Shandong University, Jinan, Shandong 250100, China
School of Mathematics and Quantitative Economics, Shandong University of Finance
and Economics, Jinan, Shandong 250014, China
E-mail address: qiujutuo@163.com
Jianguo Si
School of Mathematics, Shandong University, Jinan, Shandong 250100, China
E-mail address: sijgmath@sdu.edu.cn